Optimal. Leaf size=27 \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{4 a^3}-\frac{\text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{4 a^3} \]
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Rubi [A] time = 0.0631485, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4635, 4406, 3302} \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{4 a^3}-\frac{\text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
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Rule 4635
Rule 4406
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^2}{\sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 x}-\frac{\cos (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}\\ &=\frac{\text{Ci}\left (\sin ^{-1}(a x)\right )}{4 a^3}-\frac{\text{Ci}\left (3 \sin ^{-1}(a x)\right )}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.0060524, size = 22, normalized size = 0.81 \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )-\text{CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 22, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) }{4}}-{\frac{{\it Ci} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\arcsin \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{\arcsin \left (a x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{asin}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37149, size = 31, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Ci}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a^{3}} + \frac{\operatorname{Ci}\left (\arcsin \left (a x\right )\right )}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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